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Nature of the eigenstates in the miniband of random dimer-barrier superlattices
Superlattices and Microstructures, Vol. 30, No. 6, 2001 doi:10.1006/spmi.2002.1020 Available online at http://www.idealibrary.com on
Nature of the eigenstates in the miniband of random dimer-barrier superlattices S. B ENTATA† , B. A IT S AADI , H. S EDIKI Département de Physique, Institut du Tronc Commun, Université des Sciences et de la Technologie d’Oran (U.S.T.O.), BP 1505 El-M’nouar, 31000 Oran, Algeria (Received 17 December 2001)
The dc conductance, the universal quantum fluctuations and the resistance distribution are numerically investigated in dimer semiconductor superlattices by means of the transfer matrix formalism. We are interested in the GaAs/Alx Ga1−x As layers, having identical thickness, where the aluminium concentration x takes, at random, two different values, with the constraint that one of them appears only in pairs, i.e. the random dimer barrier (RDB). These systems exhibit a miniband of extended states, around a critical energy, lying to the typical structure of the dimer cell. The states close to this resonant energy consist of weakly localized states, while in band tails i.e. for negligible conductance, the states are strongly localized. This is evidence of the suppression of localization in the RDB superlattices. The nature of the transition between these two regimes is quantitatively investigated through relevant physical quantities. The model is, hence, clearly and statistically examined. c 2002 Elsevier Science Ltd. All rights reserved.
Key words: random dimer-barrier superlattices (RDBSL), universal conductance fluctuations (UCF), delocalized state.
1. Introduction As the Kronig–Penney model [1] provides the simplest instance of Bloch states, it has been widely used to explore the transport properties in periodic systems. This model determines analytically the band structures of the semiconductor lattices [2]. The Kronig–Penny model and its relations can easily be generalized, providing an implementation to the physics of quasi-periodic and random systems [3]. Since the original paper of Anderson [4], the problem of localization of a particle in any amount of disorder, is still of continuous interest for physicists. Mott and Twose [5] proved rigorously that all states are exponentially localized in one-dimensional full randomness parameters. Using scaling properties, the study was then extended to a higher dimension by the ‘gang of four’ [6]. Localization in all eigenstates by disorder in one-dimensional systems has often been regarded as an exact statement [7]. However, there exist several exceptions to this rule. Investigation of the mechanisms of the breaking of Anderson localization has attracted great attention. Interest in the correlation in the disorder has been † E-mail: [email protected]
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increased substantially. Evidence was found that, in the presence of internal spatial restrictions in disorder, extended states may appear at particular energies in disordered systems either with long-range correlation [24, 25] or short-range correlation [8–23]. The simplest model that exhibits the suppression of localization in disordered systems is the so-called continuous random dimer model [8, 9]: impurities are placed randomly with the main restriction to be generated with pairs, without any aggregates. The recent advances of molecular beam epitaxy (MBE) allow for an excellent control of structure parameters of semiconductors, to fabricate tailored superlattices (SL), suggesting a new type of quantum electronic device as a good candidate to carry out the metal insulator transition (MIT) experiments. This is in the aim of finding experimentally measurable magnitudes and physically realizable situations, that confirm a clear validation of the theoretical model results, about the localization or delocalization electronic states in a quasi-1D system [28–42]. Recent experiments on the study of electronic properties of GaAs/Alx Ga1−x As SL with intentional short correlated disorder by means of photoluminescence and vertical dc resistance, have already supported the existence of delocalized states in random dimer SL [26]. Such results have been predicted by DominguezAdame, in the last decade, [28–36] introducing the correlated structural disorder by means of the so-called random dimer quantum wells superlattices (RDQWSL). In such a case, the resonant tunnelling [27] appears as the principal physical mechanism, breaking down the destructive interference introduced by disorder. To the best of our knowledge very little has been done for the case of cellular disorder, namely the case for a dimer for which randomness is assumed in the height of the barriers. Up to now, only the 1D-Kronig– Penney model, especially for its sake of simplicity, has been treated, i.e. the case of a 1D array of regularly spaced δ-function potentials with paired correlated δ-function strengths [17]. Two particular kinds of Hamiltonian models have been used to show the suppression of localization, namely the tight binding model (either diagonal, off-diagonal or both) and the Kronig–Penney model in Poincaré map version. The MIT have been largely studied with some relevant magnitudes: transmission probability, Launder resistance, Lyapunov exponent, density of states, multifractal analysis, resistance (conductance) distribution, giving more detail about the suppression of localization in RDM and RDQWSL [8, 37]. Therefore, these situations, have motivated us to examine numerically, using the transfer matrix formalism [43–45], the effects of random dimmer-barrier superlattices (RDBSL) on the nature of the eigenstates of 1D-disordered SL according to the corresponding conductance distribution regime. The paper is organized as follows: the present model (RDBSL) and the mathematical formalism, which is used to calculate the conductance, are respectively presented in Section 2. For definiteness, samples, of about 2 · 104 Å length, describing the mesoscopic systems are considered. Results about conductance, universal conductance fluctuation (UCF), and resistance distribution probability are discussed in Section 3. Finally, we conclude the paper with a brief summary of the main results.
2. Formalism In this section, we study the electronic properties of the RDBSL in the stationary case. For definiteness, we consider quantum well-based SL constituted by two semiconductor materials with the same well width a and barrier thickness b in the whole sample which in turn preserves the periodicity of the lattice along the growing axis; the unit supercell having the period d = a + b. For an appropriate understanding of the RDB effect on the nature of the electronic and transport properties, the physical picture may be handled through the investigation of states close to the bottom of the conduction miniband with k⊥ = 0. As usual, the nonparabolicity effects can be neglected without loss of generality. Under these circumstances, the oneelectron one-band effective-mass Hamiltonian provides a satisfactory description: 2 h¯ d 1 d − + V (z) ψ(z) = Eψ(z). (1) SL 2 dz m ∗ (z) dz
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Here the SL potential VSL derives directly from the different energies of the conduction-band edge of the two semiconductor materials (GaAs and Alx Ga1−x As) at the interfaces. In this model of disordered SL, we consider that the height of the barriers takes at random only two values, namely V and V . These two energies are proportional to the two possible values of the Al fraction in the Alx Ga1−x As barriers, x, for x ≤ 0.45. The sequence of energies is short-range correlated since the V only appears forming pairs, e.g. V V V V V V V V V . . .. In the following treatment, we include the electron effective masses according to the different regions of the potential: m b and m b corresponding to barrier heights V and V , respectively, and m w to the well. The transmission coefficient and all the related physical quantities of interest (dc conductance, relative fluctuation of conductance, resistance and its probability distribution) at zero temperature can be conveniently computed within the framework of the transfer matrix formalism. Using the Bastard conditions of continuity [46], for an incident electron coming from the left one has the relation between the reflected and transmitted amplitude, r and t, respectively: 1 t = M(0, L) (2) r 0 a simple algebra yields the transfer matrix M(0, L) as: ik m ∗w − m ∗w −1 1 M(0, L) = − S(0, L) ik ik 2ik − m ∗ 1 m ∗w w
1 − mik∗
.
w
Here the diffusion matrix S(0, L) can be formulated in terms of the elementary diffusion matrices G j (l) associated to each region j of the potential having a width l as the product: Y S11 S12 N S(0, L) = j = 0 G j (l) = . (3) S21 S22 The transmission coefficient is then given by: 4
τ= (S11 + S22
)2
+
k m ∗w S12
−
m ∗w k S21
2 .
(4)
This expression measures the electron interaction with the structure through the elements of the diffusion 2m ∗ E matrix S(0, L) and the wavevector defined by k 2 = w2 . Once the transmission coefficient is determined, h¯ the dimensionless dc conductance is deduced by means the Landauer formula [47]: σ (E) =
τ (E) 1 − τ (E)
(5)
and then all the desired physical quantities of interest can be computed.
3. Results and discussion Within the following description, several parameters can be varied, namely: the height of the potential barriers V and V , the width of the quantum wells (QW) a, the thickness of the potential barriers b, the concentration of the dimer c in the SL and the length of the system L through the number N of supercells. Each of these parameters has a specific physical bearing: the width a determines the number of minibands while the thickness b acts on the spread of these minibands by controlling the strength of the interaction between neighbour states belonging to neighbour wells. Here the concentration of dimer c measures the degree of disorder.
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For a proper understanding, we have treated the overquoted GaAs/Alx Ga1−x As as the semiconductor SL. This material has a long and rich history and presents a great challenge for technological purposes. Moreover all the desired experimental parameters involved in our calculations are available in the literature and, besides, it serves as test for our computed magnitudes. In particular, the SL potential VSL may be expressed in terms of the aluminium concentration x in Alx Ga1−x As, using the rule 60% for the conduction-band offset, via the relation [48]: VSL (x) = {0.6(1.247x)
for (0 ≤ x ≤ 0.45)}.
(6)
This interval of x delimits the region where Alx Ga1−x As presents a direct gap in the direction 0. As well as the effective mass in this region: m(x) = (0.067 + 0.083x) · m 0
(7)
m 0 being the free electron mass. We have computed various physical quantities to discriminate the effects of the RDB on the nature of the eigenstates and their transport properties in mesoscopic disordered SL having length up to L ∼ 16 × 103 Å, i.e. N = 300. In particular, we have numerically investigated the conductance σ (E), the relative conductance fluctuations 1σ σ , the resistance ρ(E) ≈ 1/σ (E) and its probability distribution W (all these functions are taken as dimensionless). All the results reported here correspond to 104 ensemble averages especially to obtain a desired accuracy for the fit of the probability distribution of the resistance. As typical values of physical parameters we have taken a = 30 Å, b = 24 Å, V = 0.33 eV and V = 0.26 eV. We have also considered two different concentrations c = 0.3 and c = 0.4. Such choice yields one miniband lying in the wells. For convenience the bottom of the wells of GaAs serves as a reference for all the energies as usual. The effective masses are m ∗w = 0.067m 0 , m b = 0.1035m 0 and m b = 0.096m 0 corresponding, respectively, to the two aluminium concentrations x1 = 0.44 and x2 = 0.35 in Alx Ga1−x As given the barrier height V and V [48]. 3.1. Conductance For the above-mentioned conditions, we have reported in Fig. 1 the results for the conductance σ (E) versus the electron energy for different concentrations. Mainly one can observe the existence of one miniband ranging from 117 up to 181 meV within the dimer well. There are three regions in the miniband: two subminibands located at the resonant energies Er 1 = 131 meV and Er 2 = 173 meV separated by a large valley E v = 153 meV. The origin of these regions is directly related to the loss of long-range quantum coherence of the electron. Unexpectedly, the DBSL support another type of resonance, its origin being completely different. Let us take a system constructed with two kinds of blocks distributed randomly on a lattice. It is evident that the effects of randomness will be removed when, for a given electron energy, the positions of two consecutive different blocks can be interchanged [39]. In this case all the blocks of each type can be moved to one of the two sides. We can represent this process as follows: V V V V V V V V V V V → V V V V V V V V V V V . Moreover it appears that the position of the resonant energies does not depend on the degree of disorder, they are identical for both c = 0.3 and c = 0.4. The intensity of the peak associated to the resonant energy decreases with increasing the degree of disorder. One can conclude that these energies depend only on the structure of the SL, i.e. the widths of the barriers and the wells, the height of the barriers and the effective masses. This feature indicates the existence of different types of eigenstates: those having a high conductance close to the resonant energy and those with low conductance. Furthermore, the width of the miniband is sensitive to the degree of disorder and decrease with increasing the degree of disorder. Near the resonance, Fig. 1 reveals a remarkable feature: the curve of the conductance exhibits a high number of oscillations which are very close together in the regime of high conductance and low concentration and behaves rather smoothly in the band tails of the low conductance. Indeed in agreement
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Fig. 1. Conductance σ as a function of incident electron energy E for different concentrations of dimers.
with previous reports [28–36], in the absence of ensemble averages, the conductance presents several narrow peaks related to the number of the wells in the RDBSL displaying a transmittance close to l. It is apparent that even within the average procedure these peaks are reduced, they are robust enough to survive. This feature indicates the existence of different types of eigenstates: those having a high conductance close to the resonant energy and those with low conductance. The particular behaviour of σ (E) suggests the existence of a cross-over between separating the two phases: strongly localized states in the band tails from weakly localized close to the resonant energy. This finding has prompted us to investigate with particular attention the critical region of the transition by means of other physical quantities. 3.2. UCF We have studied the relative fluctuation of the conductance for the same RDBSL, defined as s 1σ hσ 2 i = −1 σ hσ i2
(8)
where the symbol h· · ·i denotes the average of the ensemble. Our numerical results for the relative fluctuations of the conductance as a function of the electron energy are presented in Fig. 2. It is mainly observed
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Fig. 2. Relative fluctuations of the conductance 1σ σ versus electron energy E for different concentrations of dimers.
in different regimes: (i) close to the resonant energies, the existence of a plateau appears over a large range of energies for which the fluctuations are severely reduced, (ii) near the band edges, the fluctuations increase and become more and more significant as the energy goes to the edges of the miniband. These fluctuations are known to be universal and inherent to any mesoscopic devices. Under the condition L < L φ quantum interference effects play a significant role, even in the presence of disorder [28]. The magnitude of the fluctuations discriminates the nature of the eigenstates. The large fluctuations are associated to the strong localized regime and the plateau corresponds to a set of weakly localized states allowing good transport properties. Therefore, the intermediate region supports the existence of a transition between the two regimes. The different behaviour of the transition should be noticed: smooth for the lower energy and abrupt for the higher one. Qualitatively this finding is in agreement with the results obtained by Berman et al. [28] where they consider the case of the RDQW. However as already noted before we should mention here that the RDB appears to be more relevant in suppressing localization than the RDQW, for the same concentration c = 0.4 the RDB reveals the existence of a plateau which spreads from E = 125 meV over a range of 1E = 54 meV, whereas in the case of RDQW the width of the plateau is about 1E = 2 meV [28]. From the above we can say that the UCF are severely reduced in the regime of weak localization. This involves that the good transport properties of RDB are quite independent of the particular realization of the system. In contrast, in the regime of strong localization, the fluctuations are very sensitive to the local environment.
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Fig. 3. Average value of the logarithm of the resistance ln ρ as a function of the length of the system L for different energies lying from the left of Er 1 (121, 123, 125, 127 meV and Er 1 = 131 meV).
3.3. Resistance For a better understanding, we have plotted, in Fig. 3, the results for the logarithm of the resistance as a function of the length of the system size for various electron energies in order to characterize the nature of the eigenstates. According to the previous observations different regimes are clearly revealed. Mainly for almost all energies, the logarithm of the resistance behaves linearly with L. This expected result suggests the well known exponential behaviour of the resistance for disordered systems with the length of the system size; i.e. ρ = ρ0 exp(L/ξ )
(9)
where ξ represents the localization length. A close inspection of Fig. 3 indicates different behaviours: (i) close to the resonant energies, the curve ln ρ is linear with very high values of ξ translating the weakly localized character of the eigenstates and their ability for transport properties, this regime applies so long as the localization length is much greater than the system size, (ii) the other regime corresponds to oscillations which become more significant with increasing L in the case of strongly localized states. Here the slope indicates a localization length much smaller than the system size, about 10 orders in magnitude. 3.4. Probability distribution For a proper knowledge of the nature of the eigenstates, we have reported the probability distribution W of ln ρ for various energies belonging to different regimes. According to these different energies the
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Fig. 4. Probability distribution W of ln ρ for the resonant energies (Er 1 = 131 meV and Er 2 = 173 meV).
probability distribution W (ln ρ, L) exhibits different behaviours: (i) for regions very close to the resonant energies W (ln ρ, L) appears to be constituted by two components (see Fig. 4). The first one corresponds to the low resistance regime and is asymmetric. The second one is associated with the high resistance but with a considerable reduced magnitude and exhibits a roughly symmetric character. This feature translates the nature of the eigenstates in this regime. Basically, these two components suggest the coexistence of two types of states. For Er 2 = 173 meV, W is very significant up to a peak at ln ρ ≈ 2.95 and then drops abruptly to zero over a large plateau from ln ρ ≈ 8.21 up to 10.56. This finding is qualitatively similar with that of Berman et al. [28] but differs quantitatively in the low resistance regime for which our probability distribution exhibits higher values translating the efficiency of the present type of dimer in suppressing localization. Above ln ρ ≈ 10.56 this value W shows a long tail in the high resistance regime with a considerable reduced magnitude. Unfortunately, we have not been able to deduce any convincing fit of the curve. Such behaviour is a signature of a weakly localized regime since the probability to find high resistances is not vanishing, (ii) as one moves in energy toward the band edge the asymmetric characters of the first component disappear and become more and more symmetric. Figure 5 shows that the gap between the two components closed corresponding to the transition, (iii) for energies in band tails, from Fig. 6 W is close to the Gaussian distribution in ln ρ (logarithmically normal in ρ). As reported by Melnicov [49] and Berman et al. [28], for a 1D-disordered system of length L s (measured in the units of the mean free path length l), for large L s and ln ρ (strong localized limit), the asymptotic of the distribution function has the
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Fig. 5. Probability distribution W of ln ρ for different energies lying from the left of Er 1 to the edge of the miniband (127 and 125 meV).
form: (L s −ln ρ)2 1 W (ln ρ, L s ) = √ e− 4L s , in the limit L s 1 and ln ρ 1. (10) 4π L s It then follows from eqn (10) that in the strong localized limit, the quantity ln ρ has a Gaussian distribution (logarithmically normal in ρ) with a variance (2L)1/2 . From the fit of the curve in Fig. 6, one can easily deduce an interesting physical parameter, namely the mean free path l ≈ 2257 Å at E = 181 meV.
4. Conclusion We have numerically examined the effects of RDB on the electronic and transport properties of 1Ddisordered SL and in particular GaAs/Alx Ga1−x As SLs. In the earlier description, we have introduced two fundamental ingredients to counteract the destructive influence of disorder: the periodicity along the growing axis and the short-range correlation. These two features create the conditions favouring tunnelling of electrons. The comparison of the effect of the RDQW [28] and the RDB on the electronic states indicates that although qualitatively similar, they act in a different way: the RDQW on the intensity of the resonance and the RDB on the width of the miniband. According to the present study, we have shown the ability of the DBSL in suppressing localization by supporting two kinds of delocalized states lying within the potential structure by properly choosing the
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Fig. 6. Fit of the probability distribution W for strongly localized states (121 and 181 meV).
parameters of the DBSL. The origins of these delocalized states are completely different: one of them is due to short-range correlations whereas the other one is due to the commuting nature of the transfer matrices describing the system at certain energies. We would like to point out that the kind of commuting delocalized states we are describing are not characteristic of the DBSL in the sense that no dimer correlations are needed at all. It means that a SL with a binary distribution of barrier heights satisfying the following correlator hVn Vm i = δnm , should exhibit the same kind of delocalized state. Furthermore, these commuting extended states seem to appear in a number of binary models, for instance the random binary Kronig–Penny model with delta barriers, the same resonant energy observed by Ishii [50] and discussed by Hilke and Flores [51]. The results obtained from the various computed quantities indicate the existence of two types of states: namely weakly and strongly localized. The probability distribution appears a relevant physical quantity. For the weak localized regime, its two components indicate the superposition of purely extended and localized states such that the mixed state remains localized with large localization length. For the strong localized regime our results confirm the analytical expression of the probability distribution of Melnicov [49]. This conclusion seems to be of universal validity since it has been reported for other types of disorders. The analytical expression of the probability distribution of the resistance deserves further investigation to explain in more detail its physical nature. This is the subject of a forthcoming paper.
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Nature of the eigenstates in the miniband of random dimer-barrier superlattices S. B ENTATA† , B. A IT S AADI , H. S EDIKI Département de Physique, Institut du Tronc Commun, Université des Sciences et de la Technologie d’Oran (U.S.T.O.), BP 1505 El-M’nouar, 31000 Oran, Algeria (Received 17 December 2001)
The dc conductance, the universal quantum fluctuations and the resistance distribution are numerically investigated in dimer semiconductor superlattices by means of the transfer matrix formalism. We are interested in the GaAs/Alx Ga1−x As layers, having identical thickness, where the aluminium concentration x takes, at random, two different values, with the constraint that one of them appears only in pairs, i.e. the random dimer barrier (RDB). These systems exhibit a miniband of extended states, around a critical energy, lying to the typical structure of the dimer cell. The states close to this resonant energy consist of weakly localized states, while in band tails i.e. for negligible conductance, the states are strongly localized. This is evidence of the suppression of localization in the RDB superlattices. The nature of the transition between these two regimes is quantitatively investigated through relevant physical quantities. The model is, hence, clearly and statistically examined. c 2002 Elsevier Science Ltd. All rights reserved.
Key words: random dimer-barrier superlattices (RDBSL), universal conductance fluctuations (UCF), delocalized state.
1. Introduction As the Kronig–Penney model [1] provides the simplest instance of Bloch states, it has been widely used to explore the transport properties in periodic systems. This model determines analytically the band structures of the semiconductor lattices [2]. The Kronig–Penny model and its relations can easily be generalized, providing an implementation to the physics of quasi-periodic and random systems [3]. Since the original paper of Anderson [4], the problem of localization of a particle in any amount of disorder, is still of continuous interest for physicists. Mott and Twose [5] proved rigorously that all states are exponentially localized in one-dimensional full randomness parameters. Using scaling properties, the study was then extended to a higher dimension by the ‘gang of four’ [6]. Localization in all eigenstates by disorder in one-dimensional systems has often been regarded as an exact statement [7]. However, there exist several exceptions to this rule. Investigation of the mechanisms of the breaking of Anderson localization has attracted great attention. Interest in the correlation in the disorder has been † E-mail: [email protected]
0749–6036/01/120297 + 12
$35.00/0
c 2002 Elsevier Science Ltd. All rights reserved.
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increased substantially. Evidence was found that, in the presence of internal spatial restrictions in disorder, extended states may appear at particular energies in disordered systems either with long-range correlation [24, 25] or short-range correlation [8–23]. The simplest model that exhibits the suppression of localization in disordered systems is the so-called continuous random dimer model [8, 9]: impurities are placed randomly with the main restriction to be generated with pairs, without any aggregates. The recent advances of molecular beam epitaxy (MBE) allow for an excellent control of structure parameters of semiconductors, to fabricate tailored superlattices (SL), suggesting a new type of quantum electronic device as a good candidate to carry out the metal insulator transition (MIT) experiments. This is in the aim of finding experimentally measurable magnitudes and physically realizable situations, that confirm a clear validation of the theoretical model results, about the localization or delocalization electronic states in a quasi-1D system [28–42]. Recent experiments on the study of electronic properties of GaAs/Alx Ga1−x As SL with intentional short correlated disorder by means of photoluminescence and vertical dc resistance, have already supported the existence of delocalized states in random dimer SL [26]. Such results have been predicted by DominguezAdame, in the last decade, [28–36] introducing the correlated structural disorder by means of the so-called random dimer quantum wells superlattices (RDQWSL). In such a case, the resonant tunnelling [27] appears as the principal physical mechanism, breaking down the destructive interference introduced by disorder. To the best of our knowledge very little has been done for the case of cellular disorder, namely the case for a dimer for which randomness is assumed in the height of the barriers. Up to now, only the 1D-Kronig– Penney model, especially for its sake of simplicity, has been treated, i.e. the case of a 1D array of regularly spaced δ-function potentials with paired correlated δ-function strengths [17]. Two particular kinds of Hamiltonian models have been used to show the suppression of localization, namely the tight binding model (either diagonal, off-diagonal or both) and the Kronig–Penney model in Poincaré map version. The MIT have been largely studied with some relevant magnitudes: transmission probability, Launder resistance, Lyapunov exponent, density of states, multifractal analysis, resistance (conductance) distribution, giving more detail about the suppression of localization in RDM and RDQWSL [8, 37]. Therefore, these situations, have motivated us to examine numerically, using the transfer matrix formalism [43–45], the effects of random dimmer-barrier superlattices (RDBSL) on the nature of the eigenstates of 1D-disordered SL according to the corresponding conductance distribution regime. The paper is organized as follows: the present model (RDBSL) and the mathematical formalism, which is used to calculate the conductance, are respectively presented in Section 2. For definiteness, samples, of about 2 · 104 Å length, describing the mesoscopic systems are considered. Results about conductance, universal conductance fluctuation (UCF), and resistance distribution probability are discussed in Section 3. Finally, we conclude the paper with a brief summary of the main results.
2. Formalism In this section, we study the electronic properties of the RDBSL in the stationary case. For definiteness, we consider quantum well-based SL constituted by two semiconductor materials with the same well width a and barrier thickness b in the whole sample which in turn preserves the periodicity of the lattice along the growing axis; the unit supercell having the period d = a + b. For an appropriate understanding of the RDB effect on the nature of the electronic and transport properties, the physical picture may be handled through the investigation of states close to the bottom of the conduction miniband with k⊥ = 0. As usual, the nonparabolicity effects can be neglected without loss of generality. Under these circumstances, the oneelectron one-band effective-mass Hamiltonian provides a satisfactory description: 2 h¯ d 1 d − + V (z) ψ(z) = Eψ(z). (1) SL 2 dz m ∗ (z) dz
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Here the SL potential VSL derives directly from the different energies of the conduction-band edge of the two semiconductor materials (GaAs and Alx Ga1−x As) at the interfaces. In this model of disordered SL, we consider that the height of the barriers takes at random only two values, namely V and V . These two energies are proportional to the two possible values of the Al fraction in the Alx Ga1−x As barriers, x, for x ≤ 0.45. The sequence of energies is short-range correlated since the V only appears forming pairs, e.g. V V V V V V V V V . . .. In the following treatment, we include the electron effective masses according to the different regions of the potential: m b and m b corresponding to barrier heights V and V , respectively, and m w to the well. The transmission coefficient and all the related physical quantities of interest (dc conductance, relative fluctuation of conductance, resistance and its probability distribution) at zero temperature can be conveniently computed within the framework of the transfer matrix formalism. Using the Bastard conditions of continuity [46], for an incident electron coming from the left one has the relation between the reflected and transmitted amplitude, r and t, respectively: 1 t = M(0, L) (2) r 0 a simple algebra yields the transfer matrix M(0, L) as: ik m ∗w − m ∗w −1 1 M(0, L) = − S(0, L) ik ik 2ik − m ∗ 1 m ∗w w
1 − mik∗
.
w
Here the diffusion matrix S(0, L) can be formulated in terms of the elementary diffusion matrices G j (l) associated to each region j of the potential having a width l as the product: Y S11 S12 N S(0, L) = j = 0 G j (l) = . (3) S21 S22 The transmission coefficient is then given by: 4
τ= (S11 + S22
)2
+
k m ∗w S12
−
m ∗w k S21
2 .
(4)
This expression measures the electron interaction with the structure through the elements of the diffusion 2m ∗ E matrix S(0, L) and the wavevector defined by k 2 = w2 . Once the transmission coefficient is determined, h¯ the dimensionless dc conductance is deduced by means the Landauer formula [47]: σ (E) =
τ (E) 1 − τ (E)
(5)
and then all the desired physical quantities of interest can be computed.
3. Results and discussion Within the following description, several parameters can be varied, namely: the height of the potential barriers V and V , the width of the quantum wells (QW) a, the thickness of the potential barriers b, the concentration of the dimer c in the SL and the length of the system L through the number N of supercells. Each of these parameters has a specific physical bearing: the width a determines the number of minibands while the thickness b acts on the spread of these minibands by controlling the strength of the interaction between neighbour states belonging to neighbour wells. Here the concentration of dimer c measures the degree of disorder.
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For a proper understanding, we have treated the overquoted GaAs/Alx Ga1−x As as the semiconductor SL. This material has a long and rich history and presents a great challenge for technological purposes. Moreover all the desired experimental parameters involved in our calculations are available in the literature and, besides, it serves as test for our computed magnitudes. In particular, the SL potential VSL may be expressed in terms of the aluminium concentration x in Alx Ga1−x As, using the rule 60% for the conduction-band offset, via the relation [48]: VSL (x) = {0.6(1.247x)
for (0 ≤ x ≤ 0.45)}.
(6)
This interval of x delimits the region where Alx Ga1−x As presents a direct gap in the direction 0. As well as the effective mass in this region: m(x) = (0.067 + 0.083x) · m 0
(7)
m 0 being the free electron mass. We have computed various physical quantities to discriminate the effects of the RDB on the nature of the eigenstates and their transport properties in mesoscopic disordered SL having length up to L ∼ 16 × 103 Å, i.e. N = 300. In particular, we have numerically investigated the conductance σ (E), the relative conductance fluctuations 1σ σ , the resistance ρ(E) ≈ 1/σ (E) and its probability distribution W (all these functions are taken as dimensionless). All the results reported here correspond to 104 ensemble averages especially to obtain a desired accuracy for the fit of the probability distribution of the resistance. As typical values of physical parameters we have taken a = 30 Å, b = 24 Å, V = 0.33 eV and V = 0.26 eV. We have also considered two different concentrations c = 0.3 and c = 0.4. Such choice yields one miniband lying in the wells. For convenience the bottom of the wells of GaAs serves as a reference for all the energies as usual. The effective masses are m ∗w = 0.067m 0 , m b = 0.1035m 0 and m b = 0.096m 0 corresponding, respectively, to the two aluminium concentrations x1 = 0.44 and x2 = 0.35 in Alx Ga1−x As given the barrier height V and V [48]. 3.1. Conductance For the above-mentioned conditions, we have reported in Fig. 1 the results for the conductance σ (E) versus the electron energy for different concentrations. Mainly one can observe the existence of one miniband ranging from 117 up to 181 meV within the dimer well. There are three regions in the miniband: two subminibands located at the resonant energies Er 1 = 131 meV and Er 2 = 173 meV separated by a large valley E v = 153 meV. The origin of these regions is directly related to the loss of long-range quantum coherence of the electron. Unexpectedly, the DBSL support another type of resonance, its origin being completely different. Let us take a system constructed with two kinds of blocks distributed randomly on a lattice. It is evident that the effects of randomness will be removed when, for a given electron energy, the positions of two consecutive different blocks can be interchanged [39]. In this case all the blocks of each type can be moved to one of the two sides. We can represent this process as follows: V V V V V V V V V V V → V V V V V V V V V V V . Moreover it appears that the position of the resonant energies does not depend on the degree of disorder, they are identical for both c = 0.3 and c = 0.4. The intensity of the peak associated to the resonant energy decreases with increasing the degree of disorder. One can conclude that these energies depend only on the structure of the SL, i.e. the widths of the barriers and the wells, the height of the barriers and the effective masses. This feature indicates the existence of different types of eigenstates: those having a high conductance close to the resonant energy and those with low conductance. Furthermore, the width of the miniband is sensitive to the degree of disorder and decrease with increasing the degree of disorder. Near the resonance, Fig. 1 reveals a remarkable feature: the curve of the conductance exhibits a high number of oscillations which are very close together in the regime of high conductance and low concentration and behaves rather smoothly in the band tails of the low conductance. Indeed in agreement
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Fig. 1. Conductance σ as a function of incident electron energy E for different concentrations of dimers.
with previous reports [28–36], in the absence of ensemble averages, the conductance presents several narrow peaks related to the number of the wells in the RDBSL displaying a transmittance close to l. It is apparent that even within the average procedure these peaks are reduced, they are robust enough to survive. This feature indicates the existence of different types of eigenstates: those having a high conductance close to the resonant energy and those with low conductance. The particular behaviour of σ (E) suggests the existence of a cross-over between separating the two phases: strongly localized states in the band tails from weakly localized close to the resonant energy. This finding has prompted us to investigate with particular attention the critical region of the transition by means of other physical quantities. 3.2. UCF We have studied the relative fluctuation of the conductance for the same RDBSL, defined as s 1σ hσ 2 i = −1 σ hσ i2
(8)
where the symbol h· · ·i denotes the average of the ensemble. Our numerical results for the relative fluctuations of the conductance as a function of the electron energy are presented in Fig. 2. It is mainly observed
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Fig. 2. Relative fluctuations of the conductance 1σ σ versus electron energy E for different concentrations of dimers.
in different regimes: (i) close to the resonant energies, the existence of a plateau appears over a large range of energies for which the fluctuations are severely reduced, (ii) near the band edges, the fluctuations increase and become more and more significant as the energy goes to the edges of the miniband. These fluctuations are known to be universal and inherent to any mesoscopic devices. Under the condition L < L φ quantum interference effects play a significant role, even in the presence of disorder [28]. The magnitude of the fluctuations discriminates the nature of the eigenstates. The large fluctuations are associated to the strong localized regime and the plateau corresponds to a set of weakly localized states allowing good transport properties. Therefore, the intermediate region supports the existence of a transition between the two regimes. The different behaviour of the transition should be noticed: smooth for the lower energy and abrupt for the higher one. Qualitatively this finding is in agreement with the results obtained by Berman et al. [28] where they consider the case of the RDQW. However as already noted before we should mention here that the RDB appears to be more relevant in suppressing localization than the RDQW, for the same concentration c = 0.4 the RDB reveals the existence of a plateau which spreads from E = 125 meV over a range of 1E = 54 meV, whereas in the case of RDQW the width of the plateau is about 1E = 2 meV [28]. From the above we can say that the UCF are severely reduced in the regime of weak localization. This involves that the good transport properties of RDB are quite independent of the particular realization of the system. In contrast, in the regime of strong localization, the fluctuations are very sensitive to the local environment.
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Fig. 3. Average value of the logarithm of the resistance ln ρ as a function of the length of the system L for different energies lying from the left of Er 1 (121, 123, 125, 127 meV and Er 1 = 131 meV).
3.3. Resistance For a better understanding, we have plotted, in Fig. 3, the results for the logarithm of the resistance as a function of the length of the system size for various electron energies in order to characterize the nature of the eigenstates. According to the previous observations different regimes are clearly revealed. Mainly for almost all energies, the logarithm of the resistance behaves linearly with L. This expected result suggests the well known exponential behaviour of the resistance for disordered systems with the length of the system size; i.e. ρ = ρ0 exp(L/ξ )
(9)
where ξ represents the localization length. A close inspection of Fig. 3 indicates different behaviours: (i) close to the resonant energies, the curve ln ρ is linear with very high values of ξ translating the weakly localized character of the eigenstates and their ability for transport properties, this regime applies so long as the localization length is much greater than the system size, (ii) the other regime corresponds to oscillations which become more significant with increasing L in the case of strongly localized states. Here the slope indicates a localization length much smaller than the system size, about 10 orders in magnitude. 3.4. Probability distribution For a proper knowledge of the nature of the eigenstates, we have reported the probability distribution W of ln ρ for various energies belonging to different regimes. According to these different energies the
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Fig. 4. Probability distribution W of ln ρ for the resonant energies (Er 1 = 131 meV and Er 2 = 173 meV).
probability distribution W (ln ρ, L) exhibits different behaviours: (i) for regions very close to the resonant energies W (ln ρ, L) appears to be constituted by two components (see Fig. 4). The first one corresponds to the low resistance regime and is asymmetric. The second one is associated with the high resistance but with a considerable reduced magnitude and exhibits a roughly symmetric character. This feature translates the nature of the eigenstates in this regime. Basically, these two components suggest the coexistence of two types of states. For Er 2 = 173 meV, W is very significant up to a peak at ln ρ ≈ 2.95 and then drops abruptly to zero over a large plateau from ln ρ ≈ 8.21 up to 10.56. This finding is qualitatively similar with that of Berman et al. [28] but differs quantitatively in the low resistance regime for which our probability distribution exhibits higher values translating the efficiency of the present type of dimer in suppressing localization. Above ln ρ ≈ 10.56 this value W shows a long tail in the high resistance regime with a considerable reduced magnitude. Unfortunately, we have not been able to deduce any convincing fit of the curve. Such behaviour is a signature of a weakly localized regime since the probability to find high resistances is not vanishing, (ii) as one moves in energy toward the band edge the asymmetric characters of the first component disappear and become more and more symmetric. Figure 5 shows that the gap between the two components closed corresponding to the transition, (iii) for energies in band tails, from Fig. 6 W is close to the Gaussian distribution in ln ρ (logarithmically normal in ρ). As reported by Melnicov [49] and Berman et al. [28], for a 1D-disordered system of length L s (measured in the units of the mean free path length l), for large L s and ln ρ (strong localized limit), the asymptotic of the distribution function has the
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Fig. 5. Probability distribution W of ln ρ for different energies lying from the left of Er 1 to the edge of the miniband (127 and 125 meV).
form: (L s −ln ρ)2 1 W (ln ρ, L s ) = √ e− 4L s , in the limit L s 1 and ln ρ 1. (10) 4π L s It then follows from eqn (10) that in the strong localized limit, the quantity ln ρ has a Gaussian distribution (logarithmically normal in ρ) with a variance (2L)1/2 . From the fit of the curve in Fig. 6, one can easily deduce an interesting physical parameter, namely the mean free path l ≈ 2257 Å at E = 181 meV.
4. Conclusion We have numerically examined the effects of RDB on the electronic and transport properties of 1Ddisordered SL and in particular GaAs/Alx Ga1−x As SLs. In the earlier description, we have introduced two fundamental ingredients to counteract the destructive influence of disorder: the periodicity along the growing axis and the short-range correlation. These two features create the conditions favouring tunnelling of electrons. The comparison of the effect of the RDQW [28] and the RDB on the electronic states indicates that although qualitatively similar, they act in a different way: the RDQW on the intensity of the resonance and the RDB on the width of the miniband. According to the present study, we have shown the ability of the DBSL in suppressing localization by supporting two kinds of delocalized states lying within the potential structure by properly choosing the
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Fig. 6. Fit of the probability distribution W for strongly localized states (121 and 181 meV).
parameters of the DBSL. The origins of these delocalized states are completely different: one of them is due to short-range correlations whereas the other one is due to the commuting nature of the transfer matrices describing the system at certain energies. We would like to point out that the kind of commuting delocalized states we are describing are not characteristic of the DBSL in the sense that no dimer correlations are needed at all. It means that a SL with a binary distribution of barrier heights satisfying the following correlator hVn Vm i = δnm , should exhibit the same kind of delocalized state. Furthermore, these commuting extended states seem to appear in a number of binary models, for instance the random binary Kronig–Penny model with delta barriers, the same resonant energy observed by Ishii [50] and discussed by Hilke and Flores [51]. The results obtained from the various computed quantities indicate the existence of two types of states: namely weakly and strongly localized. The probability distribution appears a relevant physical quantity. For the weak localized regime, its two components indicate the superposition of purely extended and localized states such that the mixed state remains localized with large localization length. For the strong localized regime our results confirm the analytical expression of the probability distribution of Melnicov [49]. This conclusion seems to be of universal validity since it has been reported for other types of disorders. The analytical expression of the probability distribution of the resistance deserves further investigation to explain in more detail its physical nature. This is the subject of a forthcoming paper.
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